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                <h1 id="1-引言"><a href="#1-引言" class="headerlink" title="1.引言"></a>1.引言</h1><p>如何像Engineer一样思考</p>
<p>举个例子：</p>
<p>假设你想要为某人准备一杯咖啡，但你并不知道这个人什么时候出现，所以你需要一个带着开关可以加热水的设备，使得水能够保持着一个合适的温度</p>
<p>我们来分析一下这个系统：</p>
<ul>
<li>系统的输入Input：电功率<ul>
<li>是通过控制开关来控制的</li>
</ul>
</li>
<li>系统的输出Output：水的温度<ul>
<li>希望保持在合适的值上</li>
</ul>
</li>
</ul>
<p>在这个系统中还有许多其他的因素在影响着，如：</p>
<blockquote>
<ul>
<li>水的质量</li>
<li>水的比热</li>
<li>空气温度</li>
<li>空气流速</li>
<li>传热效率</li>
<li>…..</li>
</ul>
</blockquote>
<p>如果我们完全掌握了这些量，那么这个系统就是完全可知的，但事实并不是如此，我们可以知道水的质量与水的比热，但是剩下的量并不能很好的获取</p>
<p>因此我们可以引入反馈系统(Feedback System)</p>
<p>引入变量E：</p>
<p>误差：Error,e            e=Twd - Tw=目标值-实际值</p>
<p>所以我们的系统加上反馈之后的电功率就变成了一个误差函数：</p>
<p>Ps = U(e)</p>
<p>那我们如何寻找U(e)呢？这个问题我们留到后面再讲</p>
<p>这里我们先来了解一些别的东西：</p>
<p>一些常用英语在系统中的含义</p>
<blockquote>
<ul>
<li>Gain→增益</li>
<li>Chaos→混沌</li>
<li>M→Mass</li>
<li>T→Temp</li>
<li>C_p→Heat capacity</li>
<li>E→Error</li>
</ul>
</blockquote>
<h1 id="2-状态空间"><a href="#2-状态空间" class="headerlink" title="2.状态空间"></a>2.状态空间</h1><p>先来看一个例子<strong>弹簧阻尼系统</strong>：</p>
<p><img src="0.png"></p>
<p>我们对质量块进行受力分析：外力f(t)、弹簧力f_k、阻尼力f_b</p>
<p>根据胡克定律：</p>
<p><img src="1.png"></p>
<p>根据牛顿第二定律：F=ma</p>
<p>我就可以得到：</p>
<p><img src="2.png"></p>
<p>在经典控制理论中，我们会用拉氏变换来找到系统的传递方程</p>
<p><img src="3.png"></p>
<p>在现代控制理论中，我们会用另一种方式也就是状态空间，状态空间可以想象成一个集合，它包含了输出、输入、状态状态变量，然后用一阶微分方程把他们表达出来</p>
<hr>
<p>回到我们的系统里，我们有一个二阶项，这显然是不对的，我们需要选择状态变量消除高阶项</p>
<p>我们选择系统的两个状态并化简系统：</p>
<p><img src="4.png"></p>
<p>写成紧凑的线性代数形式：</p>
<p><img src="5.png"></p>
<p>系统的输出y，y就是x就是z_1：</p>
<p><img src="6.png"></p>
<p>写成一般形式就是：</p>
<p><img src="7.png"></p>
<p>这就是系统的状态空间表达形式，对于我们这个系统来说：</p>
<p><img src="8.png"></p>
<p>现在来看看状态空间方程与传递函数之间的关系：</p>
<p><img src="9.png"></p>
<p>记住我们最后推导出来的公式，下面我们来将它化简：</p>
<p><img src="10.png"></p>
<p>得出的就是传递函数</p>
<p>现在来看看传递函数的分母部分，传函的分母与|SI-A|是息息相关的，令其为零即可算出一个值S,而这个S是A矩阵的特征值</p>
<p>对于传递函数来说，分母为零求出的解是它的极点，在经典控制理论中，一个传递函数的极点就决定它的稳定性，而同时A矩阵的特征值也有可能决定系统的稳定性，这点放在后面细讲</p>
<p>下面来联系一下</p>
<p>这是一个电力系统：</p>
<p><img src="11.png"></p>
<p>我们取两个节点，e1，e2</p>
<p>根据基尔霍夫电流定律：</p>
<p><img src="12.png"></p>
<p>带入化简：</p>
<p><img src="13.png"></p>
<p>这样就得到了两个状态变量对时间的倒数方程，顺便写出状态空间方程</p>
<hr>
<p>下面总结一下重点：</p>
<p><img src="14.png"></p>
<h1 id="3-相图，相轨迹"><a href="#3-相图，相轨迹" class="headerlink" title="3.相图，相轨迹"></a>3.相图，相轨迹</h1><p>这是一种分析微分方程解的方法，这里用控制的角度来理解</p>
<p>来看一个方程：</p>
<p><img src="15.png"></p>
<p>相图如下：</p>
<p><img src="16.png"></p>
<p>图中的小箭头代表x1，x2的流动，假设任取一点，最终都会跟着箭头流到一个稳定的点</p>
<p>下面主要讨论线性系统，这里放个<a target="_blank" rel="noopener" href="https://ww2.mathworks.cn/matlabcentral/fileexchange/61636-pplane">网站</a>可以用来绘制这种图，网站中还有MATLAB的插件，下载下来就可以用MATLAB来绘制这种图了</p>
<p>假设我们有一个函数：</p>
<p><img src="17.png"></p>
<p>它与x轴有两个交点</p>
<p>当你的函数起始点是在x01，x02的时候，这个x就不会有变换</p>
<p>那么我们分析一下x01，x02周围的点：</p>
<p>例如：当x&lt;0,X&gt;0时，X会向着x01移动</p>
<p><img src="18.png"></p>
<p> 在看一个栗子：</p>
<p><img src="19.png"></p>
<p>这个中的点是不稳定点</p>
<p>对于下面这个系统而言：</p>
<p><img src="20.png"></p>
<p>它的特征点有三种情况：源点、鞍点、汇点</p>
<p><img src="21.png"></p>
<p>来看一般形式，ABCD都不为零：</p>
<p><img src="22.png"></p>
<p><img src="23.png"></p>
<p>我们再来看一个例子：</p>
<p><img src="24.png"></p>
<p><img src="25.png"></p>
<p>上面第二幅图有个错误，lamda中是没有a的</p>
<p><img src="26.png"></p>
<p>在这种情况下x1，x2是在循环往复的</p>
<p>再来看个例子：</p>
<p><img src="27.png"></p>
<p>e^2it和e^-2it是椭圆，e^t是随时间不断增加的</p>
<p>所以图形是螺旋向外的，这里要搞清楚是顺时针还是逆时针</p>
<p><img src="28.png"></p>
<p>所以它的图像是逆时针的</p>
<hr>
<p>总结：</p>
<p>可以通过判断系统的特征值来判断系统：</p>
<p><img src="29.png"></p>
<p>虚部带来了什么呢？</p>
<p>虚部引进了振动：</p>
<p><img src="30.png"></p>
<p>虚部转动的方向可以通过坐标上的一个点来判断</p>
<p>对于系统来说如果想让系统稳定，它一定要lamda1，lamda2小于零或者lamda的实部小于零</p>
<h1 id="4-关于Phase-Portrait的故事"><a href="#4-关于Phase-Portrait的故事" class="headerlink" title="4.关于Phase Portrait的故事"></a>4.关于Phase Portrait的故事</h1><p>1988Steven strogatz博士后时提出来的，现在是康奈尔大学应用数学系的教授，使用的是莎士比亚笔下的人物Romeo与Juliet</p>
<p>这里使用两个中文名字，图个方便</p>
<p>男：与非            Y(t)代表与非对梦寒的爱/恨，Y&gt;0喜欢，Y&lt;0恨</p>
<p>女：梦寒            M(t)代表梦寒…                         M&gt;0…        M&lt;0…</p>
<p>这里讨论其中一种形式：</p>
<p><img src="31.png"></p>
<p>这里我们深入讨论一下：</p>
<p>这里解释一下：与非爱，梦寒恨，与非的热情减少</p>
<p><img src="32.png"></p>
<p>而当移动到零界点的时候：</p>
<p>与非恨，梦寒对与非的态度发生了转变，梦寒恨意渐渐开始减少</p>
<p><img src="33.png"></p>
<p>而又一次移动到零界点的时候：</p>
<p>梦寒爱，随着梦寒态度的转变，与非恨意逐渐减少</p>
<p><img src="34.png"></p>
<p>又一次到达临界点：</p>
<p>梦寒爱，与非由恨转爱，但是此时梦寒的热情减少</p>
<p><img src="35.png"></p>
<p>他们就这样循环往复下去，虽然你可能觉得，这也太魔幻了点，但现实往往更加魔幻，至少在数学中，你知道她(他)是爱你的，但现实生活中谁有能说得准呢？</p>
<p>回归正题我们可以发现在这个例子里：</p>
<blockquote>
<ul>
<li>1/4相爱</li>
<li>1/2一半火焰，一半冰川</li>
<li>1/4互相看不顺眼</li>
</ul>
</blockquote>
<p>其实来说这是种不错的关系，因为即使是在这1/4互看不顺眼的关系中你也可以把它理解成：**<em>离别不过是换另一种方式的陪伴，这一刻让我凝望你的眼**</em></p>
<hr>
<p>让我们来看一下另一种情况：</p>
<p><img src="36.png"></p>
<p>我们来分析一下：</p>
<p>随着时间的推移，最终二人会变成路人</p>
<p><img src="37.png"></p>
<p>我们来看看另一种情况：</p>
<p>最终他们会有两种情况，就是第一和第三象限的模样，这取决一<strong>第一映像</strong>：</p>
<p><img src="38.png"></p>
<p>当Y&gt;0与非爱着梦寒，M&lt;0梦寒讨厌与非的时候，有两种情况，一种坚持不懈终成眷属，一种半途而废终成仇人</p>
<p>我们来讨论一下这种结果：</p>
<ul>
<li><input disabled="" type="checkbox"> <strong>|a|&gt;|b|：自我意识大于对方感受，太关注自己的意识，所以最终会成为路人</strong></li>
<li><input disabled="" type="checkbox"> <strong>|a|&lt;|b|：自我意识小于对方感受，最终会有两种结果，坠入爱河和因爱生恨</strong></li>
</ul>
<p>这就验证了很多人宁愿做朋友也不愿意打破平衡</p>
<p>不认真赢不了，而认真也可能就输了</p>
<hr>
<p>总结一下：</p>
<p><img src="39.png"></p>
<p><img src="40.png"></p>
<p>拿数学去描述爱情终究是不行的，现实生活中充满了不确定性，这些不确定性才让我们感觉自己是活着的，无论这不确定是好事坏</p>
<h1 id="5-系统的可控性-Controllability"><a href="#5-系统的可控性-Controllability" class="headerlink" title="5.系统的可控性(Controllability)"></a>5.系统的可控性(Controllability)</h1><p>假设有这么一个例子：</p>
<p><img src="41.png"></p>
<p>我们可以通过控制小车的速度u</p>
<p>使得小车可以达到我们想要的位置x和x一点</p>
<p>这时候我们给它加一个拖挂，一个通过弹簧链接的车斗子</p>
<p>这时候我们是否还能通过u控制x和x一点呢？</p>
<p><img src="42.png"></p>
<p>一个系统的可控性它的定义是这样的：</p>
<p><img src="43.png"></p>
<p>简单而言：</p>
<p><img src="44.png"></p>
<p><img src="45.png"></p>
<p>也就是说，如果一系统可控，那么Co矩阵的秩一定等于n</p>
<p>来看一个例子：</p>
<p><img src="46.png"></p>
<p>来看另外一个例子：</p>
<p><img src="47.png"></p>
<p>可控性是点对点的可控还是轨迹上的可控？</p>
<p>对于这个系统，如果我们有一个x0，我们希望将他移动到xt：</p>
<p><img src="48.png"></p>
<p>那么不能直接由右至左，因为它的速度是向右的，所以我们只能不断减小速度，直到为零时施加向左方向力，让它朝左运动：</p>
<p><img src="49.png"></p>
<p>所以这其实是一个点对点的可控，我们不一定可以完全让它按照我们设定的点运动</p>
<p>我们看一下最开始的那个问题：</p>
<p><img src="50.png"></p>
<p>他就可以写成下面的形式：</p>
<p><img src="51.png"></p>
<p>我们可以用Matlab求解一下：</p>
<p><img src="52.png"></p>
<p>我们惊讶的发现，它竟然是可控的</p>
<hr>
<p>总结：</p>
<p><img src="53.png"></p>
<p>即使一个系统理论可控，但在实际系统中还需要考虑多种物理因素的约束</p>
<h1 id="6-稳定性-Stability-Lyapunou"><a href="#6-稳定性-Stability-Lyapunou" class="headerlink" title="6.稳定性(Stability-Lyapunou)"></a>6.稳定性(Stability-Lyapunou)</h1><p>我们看个例子</p>
<p><img src="54.png"></p>
<p>当三个小球离开自身位置之后，只有C球可以稳定的回到原点</p>
<p>我们观察现象之后，可以不严谨的说：稳定系统，离开平衡点后的反应随时间衰减，至少不增加….</p>
<p>A,C是稳定点，B是不稳定的</p>
<p>这里我们来看一下严谨的数学定义：</p>
<p><img src="55.png"></p>
<p>第一个是lyapunov定义的</p>
<p>第二个是渐进稳定性</p>
<p>我们来看一个例子：</p>
<p>处于小圆外大圆内的两个点</p>
<p>一个蓝色x</p>
<p>一个红色x</p>
<p>蓝色x一直处在大圆内小圆外，这叫做lyapunov稳定</p>
<p>红色x随着时间的推移回到了平衡点，这叫渐进稳定</p>
<p>对于高阶系统也一样</p>
<p>对于线性系统来说：</p>
<p><img src="56.png"></p>
<p>对于非线性系统来说：</p>
<p><img src="57.png"></p>
<p>非线性系统有两种解法，先做第一种lyapunov解法</p>
<p>第一种就是求解微分方程的解法</p>
<p>如果我们能找到一个lyapunov函数V满足上述三种条件，我们就可以将x=0称为稳定的平衡点</p>
<p>这里就可以引出两个定义</p>
<ul>
<li>PSD-Positive semi definit半正定，除了零点以外其他的点都是大于等于零的</li>
<li>NSD-Negative半负定</li>
</ul>
<p>如果是：</p>
<p><img src="58.png"></p>
<p>这里我们来看一下一个经典的模型，弹簧阻尼系统：</p>
<p><img src="59.png"></p>
<p>我们再来看一下非线性的情况：</p>
<p><img src="60.png"></p>
<p>如何寻找lyapunov函数V是一个难点</p>
<h1 id="7-线性控制器设计-Linear-Controller-Design"><a href="#7-线性控制器设计-Linear-Controller-Design" class="headerlink" title="7.线性控制器设计(Linear Controller Design)"></a>7.线性控制器设计(Linear Controller Design)</h1><p><img src="61.png"></p>
<p>Acl:闭环状态空间矩阵</p>
<p>我们可以通过调整K的值来得到我们希望Acl的特征值</p>
<p>看一个例子：</p>
<p><img src="62.png"></p>
<p><img src="63.png"></p>
<p>再来看一个例子：</p>
<p><img src="64.png"></p>
<p><img src="65.png"></p>
<p>看一个实际应用：</p>
<p><img src="66.png"></p>
<p>得到系统的动态方程</p>
<p>接下来我们可以得到状态空间方程</p>
<p><img src="67.png"></p>
<p>不可控，引入u</p>
<p><img src="68.png"></p>
<p>求特征值</p>
<p><img src="69.png"></p>
<h1 id="8-LQR控制器"><a href="#8-LQR控制器" class="headerlink" title="8.LQR控制器"></a>8.LQR控制器</h1><p><img src="70.png"></p>
<p><img src="71.png"></p>
<p><img src="72.png"></p>
<p>我们来看之前的一个例子</p>
<p><img src="73.png"></p>
<p><img src="74.png"></p>
<p>simulink操作步骤</p>
<pre><code>首先在命令行中输入参数
A矩阵
B矩阵
Q矩阵,决定响应和速度，前响应后速度
R输入大小，在这个例子里决定能耗
K=lqr(A,B,Q,R)
将得到的K顺序放入Gain中

例：
A=[0 1;10 0]
B=[0;-1]
Q=[100 0; 0 1]
R=0.01
K=lqr(A,B,Q,R)

A=[0 1;10 0]
B=[0;-1]
Q=[50 0; 0 1]
R=10
K=lqr(A,B,Q,R)
将其的到参数分别输入两个系统的Gain中</code></pre>
<p>simulink模型如下：</p>
<p><img src="81.png"></p>
<p>上</p>
<p><img src="82.png"></p>
<p>下</p>
<p><img src="83.png"></p>
<p>输出图</p>
<p><img src="84.png"></p>
<h1 id="8-5轨迹跟踪-Follow-a-Desired-Path"><a href="#8-5轨迹跟踪-Follow-a-Desired-Path" class="headerlink" title="8.5轨迹跟踪(Follow a Desired Path)"></a>8.5轨迹跟踪(Follow a Desired Path)</h1><p><img src="75.png"></p>
<p>如果我们希望小球停在五度的时候该怎么办呢</p>
<p><img src="76.png"></p>
<p><img src="78.png"></p>
<p>图中Ef=x1d而不是Ef=g/lx1d</p>
<p><img src="79.png"></p>
<p><img src="80.png"></p>
<p>simulink仿真：</p>
<p>系统框图</p>
<p><img src="85.png"></p>
<p>仿真结果</p>
<p><img src="86.png"></p>
<p>为什么说是轨迹跟踪呢，我们可以换一个输入</p>
<p>系统框图</p>
<p><img src="87.png"></p>
<p>仿真结果</p>
<p><img src="88.png"></p>
<p>我们可以看到他还是能很好的跟上设定轨迹</p>
<h1 id="9-状态观测器-Linear-Observer-Design"><a href="#9-状态观测器-Linear-Observer-Design" class="headerlink" title="9.状态观测器(Linear Observer Design)"></a>9.状态观测器(Linear Observer Design)</h1><p><img src="89.png"></p>
<p>下面介绍一下luenberger观测器，线性观测器</p>
<p><img src="90.png"></p>
<p><img src="91.png"></p>
<p>令dotx-dotx^hat=e_x</p>
<p>e_x误差，就是估计值和实际值之间的误差</p>
<p>我们希望e_x=0</p>
<p><img src="92.png"></p>
<p>举个例子</p>
<p><img src="93.png"></p>
<p><img src="94.png"></p>
<p>Simulink演示</p>
<p><img src="95.png"></p>
<p><img src="96.png"></p>
<h1 id="10-观测性与分离原理-Observability-Separation-Principle"><a href="#10-观测性与分离原理-Observability-Separation-Principle" class="headerlink" title="10.观测性与分离原理(Observability Separation Principle)"></a>10.观测性与分离原理(Observability Separation Principle)</h1><p><img src="97.png"></p>
<p><img src="98.png"></p>
<p><img src="99.png"></p>
<p>如果是非线性的可以等待后续教程</p>
<hr>
<p>参考内容：<a target="_blank" rel="noopener" href="https://www.bilibili.com/video/BV1yx411u7iX">https://www.bilibili.com/video/BV1yx411u7iX</a></p>
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                        <span class="card-title">关于FOC的学习笔记</span>
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                            永磁同步电机控制的学习笔记！
                        
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                                    电机控制
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